Now I'm not a physical chemist, but I did find this equation in my old P. Chem. book. It says that this is one measure of size, but I'm not sure how good it is relative to experimental data. I think it's only valid for one electron atoms.

average electron-nuclear separation = <r> = (a0/(2Z))*[3n^2-l(l + 1)]

Where a0 is the Bohr radius, Z is the nuclear charge, and n and l are the respective quantum numbers.

Replies so far seem to confirm what I believe is the current situation, which is that all that is known is the average radii figure for each element (found by experiment).
It seems that we rely on magnetic theory to calculate to Ionic Tables but cannot, or have not, use magnetic theory to calculate isotope radii. This seems rather peculiar since the text books tell us that the radii is determined by the magnetic force.
Given that three different methods are used to find Ionic radii and they are not in complete agreement, it seems that the statement that magnetic force determines Ionic and Isotope radii has not been fully proven beyond any doubt.
I would like to know if this is the correct state of current knowledge or have I missed something?

Actually, the radii are dependent on electrostatic potentials. Atomic, Ionic and Covalent radii are different quantities by definition. An atom/ion is not a well defined object like a tennis ball, that you can attribute an accurate radius to it. The covalent radius for a diatomic element like hydrogen or oxygen, is probably easiest specified, as it is just half the internuclear separation. Atomic radius in a solid crystalline element (like copper or silicon) is also fairly well specified in terms the lattice spacing. I think there are 2 different measures of Ionic radius, but this is not a conflict resulting from any lack of understanding. It's like the 2 different ways of specifying the length of an animal - some like to include the length of the tail, others don't.

Everything needed in this area, has been fully proven beyond any doubt.

Yes and the radius is not a smoothly periodic property. A problem that we have in comparing the radius of one element to another is which radius to choose? If you go from left to right in a periodic table, the alkali metal is the largest in the period and the halogen is the smallest. Of course then, the noble gas element is even larger than the alkali metal you started out with!

The "paradox" here is that the radii comparisons are self-inconsistent. From a logical standpoint, you should be comparing two parameters keeping all others fixed, to make a generalization such as the one made above. Secondly, it does not make much sense to compare ionic radii with covalent radii or in the case of noble gases, the vanderwall radii (which is much bigger than either of the two for the interatomic distances being measured are those between two nearest nonbonded atoms).

Yet if we compare the radii just on the basis of their values, we can make such statements like "Flourine is smaller than Lithium". You may have encountered such propositions and actually they are not wrong (as you can compare a spherical metal shell with a tennis ball though one is hard metal and is subject to different forces than the tennis ball--two completely different "worlds"). This is just the way things have been defined.

Coming back to your question, I would repeat the statement made in my first post: that there exists no single "formula" to compute the radius of an atom. Firstly, an atom has no unique boundary for you to place a marker and make a measurement--this is of course due to quantum mechanics. Secondly, all our observations are made on samples containing a large number of atoms since it is experimentally very difficult to obtain a single atom. Hence, radii measurements are conducted on samples of atoms and then averaged out to give you an estimate.

This "average" value is not really the actual value as according to quantum mechanics, the electron cloud may smear even farther than the confines of a conventional boundary surface (the orbital) it existed in. The actual value is time variant and it would be difficult for us to base conclusions about the atoms (their properties) on time variant quantities. So the average is your best bet :-), though you must understand that the orbital is a region in space where the probability of finding the electron is about 95%--so average values are not grossly incorrect for explaining a large number of propositions (and exceptions) in general chemistry.

Everything needed in this area, has been fully proven beyond any doubt.

I think the reply from maverick280857 explains why this is not the case. There is also the fact that we do not know what the various forces or mass and charge really are, or why they exist, but that is not a chemistry forum subject.
I have ordered a copy of 'Bonding and structure' by N W Alcock, but if there are other books you would recommend please let me know. My subject is 'what causes of the structure of fundamental particles and atoms to exist in their particular form'. I am somewhat puzzled by the fact that the most detailed explanation of the current teachings seems to be found in the chemistry forum and in books on chemistry, while the various physics forums and their multitude of books show little interest in anything other than predictive theories. Neither physicist or chemist seem prepared to acknowledge how little we know about structure, both being prepared to rely on predictive theories, without being unduly concerned at our lack of fundamental knowledge.

...There is also the fact that we do not know what the various forces or mass and charge really are, or why they exist, but that is not a chemistry forum subject.

Actually we do know exactly "what the various forces or mass and charge really are". Why they exist - that's a different ballgame - is not known.

I have ordered a copy of 'Bonding and structure' by N W Alcock, but if there are other books you would recommend please let me know. My subject is 'what causes of the structure of fundamental particles and atoms to exist in their particular form'.

Fundamental particles do not have structure. That is why they are called 'fundamental'. The structures of atoms, molecules and solids (crystalline and amorphous) are understood using Quantum Mechanics, and are explained completely.

I am somewhat puzzled by the fact that the most detailed explanation of the current teachings seems to be found in the chemistry forum and in books on chemistry, while the various physics forums and their multitude of books show little interest in anything other than predictive theories.

The problem of explaining the structure was solved by physicists. It is merely an application of QM. As far as physicists are concerned the questions have been answered, so there's nothing more for them to do. Chemists are more interested in structure because the chemical properties of different compounds/inorganic materials/proteins/enzymes (etc) is strongly dependent on structure. So the methods used to correlate structure and property have been developed by chemists.

Neither physicist or chemist seem prepared to acknowledge how little we know about structure, both being prepared to rely on predictive theories, without being unduly concerned at our lack of fundamental knowledge.

A whole lot (many will say 'everything') is known about the structure of atoms and molecules and solids. The structure of sub-atomic particles is a little more tricky but that too is pretty well understood.

I don't understand your complaint about "predictive theories". You do acknowledge that we can understand things that we know little about only by developing consistent theories that explain the observations that have been made? A theory gains acceptance, if it can correctly predict outcomes - if not, you throw your theory in the trash.

Extract from ELECTRODYNAMICS AND CLASSICAL THEORY OF FIELDS AND PARTICLES by A.O. BARUT, Professor of physics, University of Colorado

"But even within the framework of quantum theories the nature of the mass of the particles remains unexplained".

More recently in the introduction to The Particle Garden the author writes-
"phsysicist will not be satisfied until they know why particles have mass, charge and other particlular properties that they are observed to contain"

Extract from Wikipedia
Mathematics is widely believed to be a science, but it is not. It is more closely related to Logic; it is not a science because it makes no attempt to gain empirical knowledge. However, mathematics is the universal language of all sciences.

I wrote at length on the "Why all the nutcases forum" giving the references to confirm that what we have at present, in the opinion of leading scientist, is correctly defined as Mathematical Philosophy not Science by specialist in the field of the Socialogy of Scientific Knowledge (SSK)

Those who teach teachers how to teach are dissapointed by the atitude of those who teach particle and quantum physics in that they fail to give a balanced picture as to the true state of knowledge in their particular field.

Coming back to your question, I would repeat the statement made in my first post: that there exists no single "formula" to compute the radius of an atom. Firstly, an atom has no unique boundary for you to place a marker and make a measurement--this is of course due to quantum mechanics. Secondly, all our observations are made on samples containing a large number of atoms since it is experimentally very difficult to obtain a single atom. Hence, radii measurements are conducted on samples of atoms and then averaged out to give you an estimate.

This "average" value is not really the actual value as according to quantum mechanics, the electron cloud may smear even farther than the confines of a conventional boundary surface (the orbital) it existed in. The actual value is time variant and it would be difficult for us to base conclusions about the atoms (their properties) on time variant quantities. So the average is your best bet :-), though you must understand that the orbital is a region in space where the probability of finding the electron is about 95%--so average values are not grossly incorrect for explaining a large number of propositions (and exceptions) in general chemistry.

Just to clarify

(a) I never meant to say that things aren't understood well.
(b) I do not think we can find an "accurate" formula for the radii for two principal reasons (which are inter-related):

(1) Proceeding towards an experimental approach to determine the radii will be of NO USE as by performing any measurement, you would be changing the very parameter that you set out to measure...the radius! This is due to Heisenberg's Uncertainty Principle.

(2) Everything that you measure will be an approximation. It can NEVER be 100% accurate and so all formulae that you can "derive" or "invent" would be super approximations to the actual thing...which should in principle contain an infinite precision...an infinite number of digits after the decimal. But it can't you'd say for our realworld instruments wouldn't accept it and life would come down to a standstill--they're finite precision instruments after all. So they won't measure it, they'll round it off. The very act of rounding off a value introduces error in it.
In the mathematician's world, any error is not too small (unless he sets limits) since it is nonzero. Its sign then makes no major difference to him. To a physicist, a chemist or even a quantum mechanist, life is what he (or she) sees in the experiment that is conducted in a laboratory. Hence that "life" is finite precision and cannot lead to 100% accuracy at all.

So even if things are well understood, they are not known to the last degree (which is why this is science and not humanities). It would not only be bold but quite inaccurate to say that things are completely understood and known. For they are not, and will very likely remain so unless we invent new theories or explanations to describe facts observed in the lab which are supplementing or contradicting a seemingly inadequate theory.

So even if things are well understood, they are not known to the last degree (which is why this is science and not humanities). It would not only be bold but quite inaccurate to say that things are completely understood and known. For they are not, and will very likely remain so unless we invent new theories or explanations to describe facts observed in the lab which are supplementing or contradicting a seemingly inadequate theory.

Prediction is the only thing that is well understood hence the arguements on String theory forums.
You might claim it is a science all the references that prove you wrong are on 'Why all the nutcases' forum.
New explanations is my main interest (on Theory development), but there are two brick walls. New explanations of something already proven (my approach) attract little interest, while contradictions are unlikely to work against well proven predictive theory.
I believe that all we really need is to provide is a link between predictive theory and reality but, I am continuely criticised by those who think that that link already exists despite many statements to the contrary by leading physicists.
I can agree with nearly all the statements made about radii made by contributors to this forum. Surely, the way ahead is to explain why things are the way they are. I do not subscribe to the view that such explanation is impossible.
I will now retire to theory development before the moderators chase me away once more.

Yes, there are questions about mass that are unresolved - the mass gap from Yang-Mills is a standing problem - and the "why" of mass may never be answered. But there is no reason why you should choose the question of atomic radii as being unexplained as a result of this. For then, you might as well have said that we don't understand projectile motion, or chemical reactions, or the composition of blood. Do you have a reference that questions our understanding of atomic radii in particular, rather than ones that question our understanding of mass and charge and why the universe exists ?

and the "why" of mass may never be answered. But there is no reason why you should choose the question of atomic radii as being unexplained as a result of this.

I have been in this position several times. My project belongs on Theory development PF but, in order to get questions answered, I have to go to other forums. When queeries, like the above, arise, I have to refer to my project and PF Moderators (quite correctly) object because I am not giving the textbook answer on a site meant for students.
In reality, as you point out, there is no text book answer at present, but on my webpage http://elasticity2.tripod.com/ I have put forward a way of linking particle mass with particle radii. As particle radii are only known in crude quantities, I tried to extend my project into atomic elements where, because of the data quoted in element tables; I thought radii were well understood. I now find that is not the case at all. But the debate cannot continue on this forum because of the rules.
I have decided that I will have to look for a different way of proving my work. If I make any progress it will appear on my webpage and in Theory development.
Meanwhile many thanks for your replies, as always on these forums they are invaluable and usually end up clarifying the issue and pointing me in a new direction.